Transcript Crosby/Krieger - Idaho State University

Continuum Mechanics for Hillslopes:
Part V

Focus on constitutive relationships

Reading for next week: (Korup et al., 2012)

Homework:
◦ Translate and improve one of the lectures that
has already been given based on the reading by
Major, 2013. Add good, physical examples of
how these concepts are applied. The best of these
will be used in future course offerings. Due 26
Sept.
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Constitutive Relations

Definition:
◦ Constitutive Equations: expressions that describe the relationships
between stress and strain, or stress and rates of distortion.

Goal: to relate stress tensor to strain tensor
◦ Not derived from general laws of mechanics but rather from
empirical (laboratory or field) observations!
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Constitutive Relations

Linearly Viscous Fluid
◦ Rates of strain are related linearly to stresses
◦ Strains immediately and indefinitely upon application of shear stress
◦ Non-newtonian fluids: non-linear relationship or threshold behavior

Linearly Elastic Material
◦ Strains immediately upon application of a stress
◦ Once the stress is removed, all the strain is recovered (Think “spring”)
◦ Applicable to small elastic deformation of materials
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Travel of seismic waves through rock
Flexure of lithospheric plates under large loads
Strains are linearly related to stresses
Relationships between stress and normal strain
Relationships between shear stress and shear strain
Relationship between pressure and dilatation
“Hookian” elastic solid
Plasticity – the Coulomb Failure Rule
◦ No strain occurs until reaching a threshold (yield stress).
◦ Once crossed, the material strains indefinitely. (Think, “block on an inclined
plane”)
◦ Not proportional to stress and not recoverable after stress is removed.
Linearly Elastic Material:
Relationships between stress and normal strain

An ideal linearly elastic material:
◦ stress is linearly proportional to strain
 (Hooke’s Law)
◦ for uniaxial normal stress:
 σ represents a uniaxial normal stress (positive in
tension)
 ε represents strain (positive in elongation, - for comp.)
 E is the constant of proportionality: Young’s modulus
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Linearly Elastic Material:
Relationships between stress and normal strain
If a tensional normal
stress is exerted in the
y-coordinate direction,
Hooke’s law can be
written as:
The ratio of contraction
to extension is known as
Poisson’s ratio and is
expressed for uniaxial
stress as:
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Linearly Elastic Material:
Relationships between stress and normal strain
• Elongation in the y-direction also
causes contraction in the z-direction.
• For an isotropic material, under
uniaxial stress, the amount of
contraction in the z-direction is
identical to the amount of
contraction in the x-direction.
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• If the material is incompressible,
thus conserving volume, dilation is
zero:
z
θ = εxx+ εyy+ εzz = 0
Linearly Elastic Material:
Relationships between stress and normal strain
For an isotropic material subject to triaxial rather
than uniaxial stress, we can write the normal strains
along the coordinate directions in terms of Young’s
modulus and Poisson’s ratio as:
Using:
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Linearly Elastic Material:
Relationships between stress and normal strain
We see that normal strains in any single coordinate
direction are related to normal stresses applied in all
coordinate directions,
not just to the normal stress applied in the
direction coincident with the normal strain.
These relations can also be written in terms of principal
stresses and principal strains (1,2,3 instead of xx, yy, zz):
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Linearly Elastic Material:
Relationships between shear stress and shear strain
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•
Relationships among shear
stress, shear strain, and elastic
moduli are derived by
considering a special case of
plane stress that is known as
pure shear
•
For this case,
shall assume
•
Furthermore, we shall
consider the strain that occurs
within a plane in a coordinate
system that is rotated with
respect to the principal stress
axes.
•
Under these constraints :
, and we
.
Linearly Elastic Material:
Relationships between shear stress and shear strain
-considering this special case of
plane stress called pure shear.
applying:
to:
we get:
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Linearly Elastic Material:
Relationships between shear stress and shear strain
• Given:
• Maximum shear stress occurs on a plane
that makes a 45° angle to the principal
stress axes
• If
• Then
• Thus from
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we get:
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Linearly Viscous Fluid:
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Linearly Viscous Fluid:
For an isotropic, incompressible fluid eqn [99] can be
generalized to three dimensions as the the viscous
stress Tensor:
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Linearly Viscous Fluid:
For an isotropic, incompressible, linearly
viscous fluid, we can write the above as:
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Linearly Viscous Fluid:
If the viscosity of the fluid is constant:
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Linearly Viscous Fluid:
For an incompressible fluid, conservation of mass
dictates that the bracketed third term on the right hand
side of the equation, known as the divergence of the
velocity field, is zero.
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This expression is the x-direction component of
an equation known as the ‘Navier–Stokes
equation’ for an incompressible, linearly viscous
fluid of constant viscosity.
Linearly Viscous Fluid:
The viscous stress term on the right-hand side of eqn
[105] can be written more compactly using the Laplacian
operator
as:
The Navier–Stokes equation, derived using the conservation of
momentum equation, is a control volume representation of
Newton’s second law for an incompressible, linearly viscous
fluid. Therefore, eqn [107] states that the inertial acceleration
(left-hand term) equals the sum of forces (right-hand terms)
acting on the system. Thus, the left-hand term can be viewed as an
inertial force, and the right-hand terms as pressure, viscous, and
body forces, respectively.
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Linearly Viscous Fluid:
If we now introduce the concept of the Reynolds number (Re),
which represents the ratio of inertial forces to viscous forces in a
fluid, we can highlight some limiting applications of the Navier–
Stokes equation. For low- Re flows, viscous forces greatly exceed
inertial forces; thus, the left-hand term is negligibly small and eqn
[107] can be simplified as:
For high-Re flows viscous forces are assumed to be relatively
unimportant, and eqn [107] can be simplified to: which is known as
Euler’s equation. This equation, which chiefly balances inertial and
pressure forces, has applications, for example, in river hydraulics.
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Linearly Elastic Material:
Relationships between shear stress and shear strain
• Because we find that:
• We learn that for
• Thus for rotated coodinates:
• Giving:
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(And owing to symmetry of
shear stresses and shear strains,
εxy = εyx)
Linearly Elastic Material:
Relationships between shear stress and shear strain
• Noting that εxy is the average shear strain in the xy plane
• and is equal to ½γxy
•
(where γxy is the total, or engineering, shear strain in the xy plane)
Or:
Where:
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(the rigidity modulus, or shear modulus)
Plasticity – The Coulomb Failure Rule
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Example Application

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No time for this in this lecture.
Conservation of Momentum and
Stress Equilibrium
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Effective Stress and Effective Stress
Equilibrium
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Effective Stress and Elastic Strain
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Displacement Formulation of
Constitutive Relations
and Groundwater Flow
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